A student seeks help with two separate questions: proving that one polynomial divides
another; and determining the integer values of a function given a product of its
variables. Doctor Vogler invokes modular arithmetic to crack the proof, and attacks
the function as a quadratic polynomial.

An n-dragon is a set of n consecutive positive integers. The first
two-thirds of them is called the tail, the remaining one-third the
head, and the sum off the numbers in the tail is equal to the sum of
the numbers in the head. Find the sum of the tail of a 99,999-dragon.

In class we are shown how to square both sides of an equation or take
the square root of both sides, but is there a rule like the addition
property of equality that formally says those are valid steps?